College
Probability Theory
Learn the mathematics of randomness—from probability models to the Central Limit Theorem. Build models, experiment, derive results, and solve university-level problems through a semester of active learning. Recommended: single-variable and multivariable calculus, algebra, infinite sums, and some exposure to mathematical reasoning. AP Statistics is not a prerequisite.
Probability Theory
College · Upper Division · Advanced
The mathematics of randomness, built one idea at a time.
Construct probability models. Map outcomes to random variables. Derive distributions and expectations. Use calculus to analyze joint uncertainty, then explain why averages stabilize and standardized sums approach a normal law.
Your mathematical toolkit
Single-variable and multivariable calculus, algebra, infinite sums, and some experience with mathematical reasoning. Unit 0 refreshes the essentials. AP Statistics is not a prerequisite.
A semester of active learning
Plan for about 120 hours including reading, experiments, proofs, practice, and assessments. Work on paper alongside the labs. Random walks and Markov chains are optional enrichment.
Course journey
Follow the ideas as they grow
17 units, in the order they build on each other. Each unit's question is the thing it exists to answer.
0. Mathematical toolkit
Rebuild the language and mathematical tools used throughout a proof-based probability course: sets, sums, counting, functions, calculus, and proof.
1. Probability framework and counting
Construct probability spaces, derive the consequences of the axioms, and choose counting models whose outcomes match the experiment.
2. Conditional probability and Bayesian reasoning
Treat information as a change of reference population, combine paths and partitions, and update hypotheses with explicit likelihood and selection assumptions.
3. Independence and its limits
Distinguish disjointness, pairwise independence, mutual independence, and conditional independence using exact models and counterexamples.
4. Discrete random variables and distributions
Move from outcomes to numerical summaries, build pmfs and cdfs, and derive the principal finite and countable probability laws.
5. Expectation, indicators, and variance
Define expectation with its existence conditions, use transformations and indicators, and derive moments that quantify location and spread.
6. Continuous Random Variables
Probability densities, continuous moments, standard families, quantiles, and elementary transformations.
7. Joint Distributions
Joint supports, marginals, conditionals, independence, and expectations of pairs.
8. Covariance and Sums
Linear combinations, coupling, aggregate counts, and the effect of shared randomness.
9. Generating Functions
Probability and moment generating functions, factorial moments, transform domains, and random sums.
10. Transformations and Convolution
CDF methods, multiple branches, bivariate Jacobians, and discrete and continuous convolution.
11. Conditional Expectation
Conditional means as random variables, nested information, total variance, hierarchical models, and prediction.
12. Convergence of Random Variables
Probability and distribution limits, moment inequalities, and counterexamples that make the quantifiers matter.
13. The Law of Large Numbers
A rigorous finite-variance weak law, simulation design, Monte Carlo estimation, and the limits of averaging.
14. The Central Limit Theorem
Centering and scaling, corrected normal approximations, sample-size planning, and the boundary between ordinary fluctuations and rare events.
15. Large Deviations and Rare Events
Exponential Markov bounds, optimized rate functions, exact-versus-normal comparisons, and reliability budgets.
16. Enrichment: Random Walks and Markov Chains · Enrichment
Nearest-neighbor walks, hitting probabilities, transition matrices, stationary laws, and convergence conditions.